ALFABETO GREGO ayúscula Minúscula Nombre M Equivalente

Romano 1 Α α Alfa A 2 Β β Beta B 3 Γ γ Gamma G 4 ∆ δ Delta D 5 Ε ε Epsilon E 6 Ζ ζ Zeta Z 7 Η η Eta H 8 Θ θ ϑ Teta Q 9 Ι ι Iota I 10 Κ κ Kappa K 11 Λ λ Lambda L 12 Μ µ Mu M 13 Ν ν Nu N 14 Ξ ξ Xi X 15 Ο ο Omicron O 16 Π π ϖ Pi P 17 Ρ ρ Rho R 18 Σ σ ς Sigma S 19 Τ τ Tau T 20 Υ υ Ipsilon U 21 Φ φ ϕ Phi F 22 Χ χ Ji C 23 Ψ ψ Psi Y 24 Ω ω Omega W

NOTACIÃO

Seno. sin

cos Coseno.

tg Tangente.

sec Secante.

csc Cosecante.

ctg Cotangente.

vers Verso seno.

arcsin sinθ θ= Arco seno de un ángulo θ .

( )u f x=

sinh Seno hiperbólico.

cosh Coseno hiperbólico.

tgh Tangente hiperbólica.

ctgh Cotangente hiperbólica.

sech Secante hiperbólica.

csch Cosecante hiperbólica.

, ,u v w Funciones de x , , . ( )u u x= ( )v v x=

Conjunto de los números reales.

Conjunto de enteros. , 2, 1,0,1,2,= − −… …

Conjunto de números racionales.

c Conjunto de números irracionales.

1,2,3,= … Conjunto de números naturales.

Conjunto de números complejos.

VALOR ABSOLUTO

1 1

1 1

si 0

si 0

y

0 y 0 0

ó

ó

n n

k kk k

n n

k kk k

a aa

a a

a a

a a a a

a a a

ab a b a a

a b a b a a

= =

= =

≥⎧= ⎨− <⎩= −

≤ − ≤

≥ = ⇔ =

= =

+ ≤ + ≤

∏ ∏

∑ ∑

EXPOENTES

( )( )

/

p q p q

pp q

q

qp pq

p p p

p p

p

qp q p

a a a

aa

a

a a

a b a b

a a

b b

a a

+

−

⋅ =

=

=

⋅ = ⋅

⎛ ⎞ =⎜ ⎟⎝ ⎠

=

LOGARITMOS

10

log

log log log

log log log

log log

log lnlog

log ln

log log y log ln

xa

a a a

a a a

ra a

ba

b

e

N x a

MN M N

MM N

N

N r N

N NN

a a

N

N N N N

= ⇒

= +

= −

=

= =

= =

=

ALGUNS PRODUTOS

ad+( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( )( )( )

2 2

2 2 2

2 2 2

2

2

3 3 2 2 3

3 3 2 2 3

2 2 2 2

2

2

3 3

3 3

2 2 2

a c d ac

a b a b a b

a b a b a b a ab b

a b a b a b a ab b

x b x d x b d x bd

ax b cx d acx ad bc x bd

a b c d ac ad bc bd

a b a a b ab b

a b a a b ab b

a b c a b c ab ac bc

⋅ + =

+ ⋅ − = −

+ ⋅ + = + = + +

− ⋅ − = − = − +

+ ⋅ + = + + +

+ ⋅ + = + + +

+ ⋅ + = + + +

+ = + + +

− = − + −

+ + = + + + + +

1

1

n

n k k n n

k

a b a ab b a b

a b a a b ab b a b

a b a a b a b ab b a b

a b a b a b n− −

=

− ⋅ + + = −

− ⋅ + + + = −

− ⋅ + + + + = −

⎛ ⎞− ⋅ = − ∀ ∈⎜ ⎟⎝ ⎠∑

( ) ( )( ) ( )( ) ( )

( )

2 2 3 3

3 2 2 3 4 4

4 3 2 2 3 4 5 5

( ) ( )( ) ( )( ) ( )( ) ( )

2 2 3 3

3 2 2 3 4 4

4 3 2 2 3 4 5 5

5 4 3 2 2 3 4 5 6 6

a b a ab b a b

a b a a b ab b a b

a b a a b a b ab b a b

a b a a b a b a b ab b a b

+ ⋅ − + = +

+ ⋅ − + − = −

+ ⋅ − + − + = +

+ ⋅ − + − + − = −

( ) ( )

( ) ( )

1 1

1

1 1

1

1 impar

1 par

nk n k k n n

k

nk n k k n n

k

a b a b a b n

a b a b a b n

+ − −

=

+ − −

=

⎛ ⎞+ ⋅ − = + ∀ ∈⎜ ⎟⎝ ⎠⎛ ⎞+ ⋅ − = − ∀ ∈⎜ ⎟⎝ ⎠

∑

∑

CONSTANTES

9…3.1415926535

2.71828182846e

π == …

TRIGONOMETRÍA

1sen csc

sen1

cos seccos

sen 1tg ctg

cos tg

CO

HIPCA

HIPCO

CA

θ θθ

θ θθ

θθ θθ θ

= =

= =

= = =

radianes=180π

CA

COHIP

θ

θ sin cos tg ctg sec csc

0 0 1 0 ∞ 1 ∞30 1 2 3 2 1 3 3 2 3 245 1 2 1 2 1 1 2 2

60 3 2 1 2 3 1 3 2 2 390 1 0 ∞ 0 ∞ 1

[ ]

[ ]

sin ,2 2

cos 0,

tg ,2 2

1ctg tg 0,

1sec cos 0,

1csc sen ,

2 2

y x y

y x y

y x y

y x yx

y x yx

y x yx

π π

π

π π

π

π

π π

⎡ ⎤= ∠ ∈ −⎢ ⎥⎣ ⎦= ∠ ∈

= ∠ ∈ −

= ∠ = ∠ ∈

= ∠ = ∠ ∈

⎡ ⎤= ∠ = ∠ ∈ −⎢ ⎥⎣ ⎦

IDENTIDADES TRIGONOMÉTRICAS

2 2

2 2

2 2

sin cos 1

1 ctg csc

tg 1 sec

θ θθ θ

θ θ

+ =

+ =

+ =

( )( )( )

sin sin

cos cos

tg tg

θ θ

θ θ

θ θ

− = −

− =

− = −

( )( )( )( )( )( )( ) ( )( ) ( )( )

sin 2 sin

cos 2 cos

tg 2 tg

sin sin

cos cos

tg tg

sin 1 sin

cos 1 cos

tg tg

n

n

n

n

n

θ π θ

θ π θ

θ π θ

θ π θ

θ π θ

θ π θ

θ π θ

θ π θ

θ π θ

+ =

+ =

+ =

+ = −

+ = −

+ =

+ = −

+ = −

+ =

( )( ) ( )( )

( )

sin 0

cos 1

tg 0

2 1sin 1

2

2 1cos 0

2

2 1tg

2

n

n

n

n

n

n

n

n

π

π

π

π

π

π

=

= −

=

+⎛ ⎞ = −⎜ ⎟⎝ ⎠

+⎛ ⎞ =⎜ ⎟⎝ ⎠

+⎛ ⎞ = ∞⎜ ⎟⎝ ⎠

SUMAS Y PRODUCTOS n

( )

( )

1 21

1

1 1

1 1 1

1 0

n kk

n

k

n n

k kk k

n n n

k k k kk k k

n

k k nk

a a a a

c nc

ca c a

a b a b

a a a a

=

=

= =

= = =

−=

+ + + =

=

=

+ = +

− = −

∑

∑

∑ ∑

∑ ∑ ∑

∑

( )

1

( )

( )

( )

( )

( )

( )( )

1

1

1

2

1

2 3 2

1

3 4 3 2

1

4 5 4 3

1

2

1

1 2 12

=2

1

1 1

1

2

12 3

6

12

4

16 15 10

30

1 3 5 2 1

!

n

k

nnk

k

n

k

n

k

n

k

n

k

n

k

na k d a n d

na l

r a rlar a

r r

k n n

k n n n

k n n n

k n n n n

n n

n k

n n

k

=

−

=

=

=

=

=

=

+ − = + −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

+

− −= =

− −

= +

= + +

= + +

= + + −

+ + + + − =

=

⎛ ⎞=⎜ ⎟

⎝ ⎠

∑

∑

∑

∑

∑

∑

∏

( )

( )0

!,

! !

nn n k k

k

k nn k k

nx y x y

k−

=

≤−

⎛ ⎞+ = ⎜ ⎟

⎝ ⎠∑

( ) 1 21 2 1 2

1 2

!

! ! !k

n nn nk k

k

nx x x x x x

n n n+ + + = ⋅∑

sin cos2

cos sin2

πθ θ

πθ θ

⎛ ⎞= −⎜ ⎟⎝ ⎠⎛ ⎞= +⎜ ⎟⎝ ⎠

( )( )

( )

( )

( )

2 2

2

2

2

2

sin sin cos cos sin

cos cos cos sin sin

tg tgtg

1 tg tg

sin 2 2sin cos

cos 2 cos sin

2 tgtg 2

1 tg

1sin 1 cos 2

21

cos 1 cos 22

1 cos 2tg

1 cos 2

α β α β α β

α β α β α βα βα βα β

θ θ θθ θ θ

θθθ

θ θ

θ θ

θθθ

± = ±

± =

±± =

=

= −

=−

= −

= +

−=

+

∓

∓

( ) ( )

( ) ( )

( ) ( )

( ) ( )

1 1sin sin 2sin cos

2 21 1

sin sin 2sin cos2 21 1

cos cos 2cos cos2 21 1

cos cos 2sin sin2 2

α β α β α β

α β α β α β

α β α β α β

α β α β α β

+ = + ⋅ −

− = − ⋅ +

+ = + ⋅ −

− = − + ⋅ −

( )sintg tg

cos cos

α βα β

α β±

± =⋅

( ) ( )

( ) ( )

( ) ( )

1sin cos sin sin

21

sin sin cos cos21

cos cos cos cos2

α β α β α β

α β α β α β

α β α β α β

⋅ = − + +⎡ ⎤⎣ ⎦

⋅ = − − +⎡ ⎤⎣ ⎦

⋅ = − + +⎡ ⎤⎣ ⎦

tg tgtg tg

ctg ctg

α βα βα β+

⋅ =+

FUNCIONES HIPERBÓLICAS

sinh2

cosh2

sinhtgh

cosh

1ctgh

tgh

1 2sech

cosh1 2

cschsinh

x x

x x

x

x

x x

x x

x x

x

e ex

x e ex

x e e

e ex

x e e

xx e e

xx e e

−

−

−

−

−

−

−

=

+=

−= =

++

= =−

= =+

= =−

x xe e−−

x

x

[

]

sinh :

cosh : 1,

tgh : 1,1

ctgh : 0 , 1 1,

sech : 0,1

csch : 0 0

→

→ ∞

→ −

− → −∞ − ∪ ∞

→

− → −

FUNCIONES HIPERBÓLICAS INV

( )( )

1 2

1 2

1

1

21

21

sinh ln 1 ,

cosh ln 1 , 1

1 1tgh ln , 1

2 1

1 1ctgh ln , 1

2 1

1 1sech ln , 0 1

1 1csch ln , 0

x x x x

x x x x

xx x

x

xx x

x

xx x

x

xx x

x x

−

−

−

−

−

−

= + + ∀ ∈

= ± − ≥

+⎛ ⎞= <⎜ ⎟−⎝ ⎠+⎛ ⎞= >⎜ ⎟−⎝ ⎠

⎛ ⎞± −⎜ ⎟= < ≤⎜ ⎟⎝ ⎠⎛ ⎞+⎜ ⎟= + ≠⎜ ⎟⎝ ⎠

IDENTIDADES DE FUNCS HIP 2 2sinh 1x x

( )( )( )

2 2

2 2

cosh

1 tgh sech

ctgh 1 csch

sinh sinh

cosh cosh

tgh tgh

x x

x x

x x

x x

x x

− =

− =

− = −

− =

− = −

− =

( )( )

( )

2 2

2

sinh sinh cosh cosh sinh

cosh cosh cosh sinh sinh

tgh tghtgh

1 tgh tgh

sinh 2 2sinh cosh

cosh 2 cosh sinh

2 tghtgh 2

1 tgh

x y x y x y

x y x y x y

x yx y

x y

x x x

x x x

xx

x

± = ±

± = ±

±± =

±=

= +

=+

( )

( )

2

2

2

1sinh cosh 2 1

21

cosh cosh 2 12

cosh 2 1tgh

cosh 2 1

x x

x x

xx

x

= −

= +

−=

+

sinh 2tgh

cosh 2 1

xx

x=

+

cosh sinh

cosh sinh

x

x

e x x

e x x−

= +

= − OTRAS

( ) ( )

2

2

2

0

4

2

4 discriminante

exp cos sin si ,

ax bx c

b b acx

a

b ac

i e iαα β β β α β

+ + =

− ± −⇒ =

− =

± = ± ∈

LÍMITES

( )1

0

0

0

0

1

lim 1 2.71828...

1lim 1

senlim 1

1 coslim 0

1lim 1

1lim 1

ln

xx

x

x

x

x

x

x

x

x e

ex

x

xx

x

e

xx

x

→

→∞

→

→

→

→

+ = =

⎛ ⎞+ =⎜ ⎟⎝ ⎠

=

−=

−=

−=

DERIVADAS

( ) ( ) ( )

( )

( )

( )

( )

( )

0 0

1

lim lim

0

xx x

n n

f x x f xdf yD f x

dx x xd

cdxd

cx cdxd

cx ncxdxd du dv dw

u v wdx dx dx dxd du

cu cdx dx

∆ → ∆ →

−

+ ∆ − ∆= = =

∆ ∆

=

=

=

± ± ± = ± ± ±

=

( )

( )

( ) ( )

( )

2

1n n

d dv duuv u v

dx dx dxd dw dv du

uvw uv uw vwdx dx dx dx

v du dx u dv dxd u

dx v v

d duu nu

dx dx−

= +

= + +

−⎛ ⎞ =⎜ ⎟⎝ ⎠

=

( )( )

( )( )

12

1 2

(Regla de la Cadena)

1

donde

dF dF du

dx du dxdu

dx dx du

dF dudF

dx dx du

x f tf tdy dtdy

dx dx dt f t y f t

= ⋅

=

=

=⎧′ ⎪= = ⎨′ =⎪⎩

DERIVADA DE FUNCS LOG & EXP

( )

( )

( )

( )

( )

( ) 1

ln

loglog

loglog 0, 1

ln

ln

aa

u u

u u

v v v

udx u u dxd e du

udx u dx

ed duu a

dx u dxd du

e edx dxd du

a a adx dxd du dv

u vu u udx dx dx

−

= = ⋅

= ⋅

= ⋅ >

= ⋅

= ⋅

= + ⋅ ⋅

1d du dx du

a ≠

DERIVADA DE FUNCIONES TRIGO

( )

( )

( )

( )

( )

( )

( )

2

2

sin cos

cos sin

tg sec

ctg csc

sec sec tg

csc csc ctg

vers sen

u udx dxd du

u udx dxd du

u udx dxd du

u udx dxd du

u u udx dxd du

u u udx dxd du

u udx dx

=

= −

=

= −

=

= −

=

d du

DERIVADA DE FUNCS HIPERBÓLICAS

2

2

sinh cosh

cosh sinh

tgh sech

ctgh csch

sech sech tgh

csch csch ctgh

u udx dxd du

u udx dxd du

u udx dxd du

u udx dxd du

u u udx dxd du

u u udx dx

=

=

=

= −

= −

= −

d du

DERIV DE FUNCS TRIGO INVER

( )

( )

( )

( )

( )

( )

( )

2

2

2

2

2

2

2

sin1

1cos

11

tg1

1ctg

1si 11

secsi 11

si 11csc

si 111

vers2

udx dxud du

udx dxud du

udx dxud du

udx dxu

ud duu

udx dxu u

ud duu

udx dxu ud du

udx dxu u

∠ = ⋅−

∠ = − ⋅−

∠ = ⋅+

∠ = − ⋅+

+ >⎧∠ = ± ⋅ ⎨

1d du

− < −⎩−− >⎧

∠ = ⋅ ⎨+ < −⎩−

∠ = ⋅−

∓

DERIVADA DE FUNCS HIP INV

1

2

-11

-12

12

12

1

1

12

senh1

si cosh 01cosh , 1

si cosh 01

1tgh , 1

11

ctgh , 11

si sech 0, 0,11sech

si sech 0, 0,11

udx dxu

ud duu u

dx dx uu

d duu u

dx u dxd du

u udx u dx

u ud duu

dx dx u uu u

−

−

−

−

−−

−

= ⋅+

⎧+ >± ⎪= ⋅ > ⎨− <− ⎪⎩

= ⋅ <−

= ⋅ >−

⎧− > ∈⎪= ⋅ ⎨+ < ∈− ⎩

∓

1d du

1

2

1csch , 0

1

d duu u

dx dxu u

−

⎪

= − ⋅ ≠+

INTEGRALES DEFINIDAS, PROPIEDADES Nota. Para todas las fórmulas de integración deberá agregarse una constante arbitraria c (constante de

integración).

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( )

( )

( ) ( ) ( )( ) [ ]

( ) ( )( ) ( ) [ ]

( ) ( )

0

, , ,

,

si

b b b

a a a

b b

a a

b c b

a a c

b a

a b

a

a

b

a

b b

a a

b b

a a

f x g x dx f x dx g x dx

cf x dx c f x dx c

f x dx f x dx f x dx

f x dx f x dx

f x dx

m b a f x dx M b a

m f x M x a b m M

f x dx g x dx

f x g x x a b

f x dx f x dx a b

± = ±

= ⋅ ∈

= +

= −

=

⋅ − ≤ ≤ ⋅ −

⇔ ≤ ≤ ∀ ∈ ∈

≤

⇔ ≤ ∀ ∈

≤ <

∫ ∫ ∫∫ ∫∫ ∫ ∫∫ ∫∫

∫

∫ ∫

∫ ∫

INTEGRALES

( ) ( )

( )

( )1

Integración por partes

11

ln

nn

adx ax

af x dx a f x dx

u v w dx udx vdx wdx

udv uv vdu

uu du n

ndu

uu

+

=

=

± ± ± = ± ± ±

= −

= ≠ −+

=

∫∫ ∫∫ ∫ ∫ ∫∫ ∫

∫

∫

INTEGRALES DE FUNCS LOG & EXP

( )

( )

( ) ( )

( )

( )

2

2

0

1ln

1

ln ln

1

ln ln ln 1

1log ln ln 1

ln ln

log 2log 14

ln 2ln 14

u u

uu

uu

u u

a

a a

e du e

aaa du

aa

aua du u

a a

ue du e u

udu u u u u u

uudu u u u u

a a

uu udu u

uu udu u

=

>⎧= ⎨ ≠⎩

⎛ ⎞= ⋅ −⎜ ⎟⎝ ⎠

= −

= − = −

= − = −

= ⋅ −

= −

∫

∫

∫

∫∫

∫

∫

∫

INTEGRALES DE FUNCS TRIGO

2

2

sin cos

cos sin

sec tg

csc ctg

sec tg sec

csc ctg csc

udu u

udu u

udu u

udu u

u udu u

u udu u

= −

=

=

= −

=

= −

∫∫∫∫∫∫

tg ln cos ln sec

ctg ln sin

sec ln sec tg

csc ln csc ctg

udu u u

udu u

udu u u

udu u u

= − =

=

= +

= −

∫∫∫∫

( )

2

2

2

2

1sin sin 2

2 41

cos sin 22 4

tg tg

ctg ctg

uudu u

uudu u

udu u u

udu u u

= −

= +

= −

= − +

∫

∫∫∫

sin sin cos

cos cos sin

u udu u u u

u udu u u u

= −

= +

∫∫

INTEGRALES DE FUNCS TRIGO INV

( )

( )

2

2

2

2

2

2

sin sin 1

cos cos 1

tg tg ln 1

ctg ctg ln 1

sec sec ln 1

sec cosh

csc csc ln 1

csc cosh

udu u u u

udu u u u

udu u u u

udu u u u

udu u u u u

u u u

udu u u u u

u u u

∠ = ∠ + −

∠ = ∠ − −

∠ = ∠ − +

∠ = ∠ + +

∠ = ∠ − +

= ∠ −∠

∠ = ∠ + + −

= ∠ +∠

∫∫∫∫∫

∫

−

INTEGRALES DE FUNCS HIP

2

2

sinh cosh

cosh sinh

sech tgh

csch ctgh

sech tgh sech

csch ctgh csch

udu u

udu u

udu u

udu u

u udu u

u udu u

=

=

=

= −

= −

= −

∫∫∫∫∫∫

( )( )1

tgh ln cosh

ctgh ln sinh

sech tg sinh

csch ctgh cosh

1 ln tgh

2

udu u

udu u

udu u

udu u

u

−

=

=

= ∠

= −

=

∫∫∫∫

INTEGRALES DE FRAC

( )

( )

2 2

2 22 2

2 22 2

tg

1 ctg

1ln

21

ln 2

du

u a a au

a adu u a

u au a a u a

du a uu a

a u a a u

= ∠+

= − ∠

−= >

− ++

= <− −

∫

∫

∫

1 u

INTEGRALES CON

( )

( )

2 2

2 2

2 2

2 2 2 2

2 2

22 2 2 2

22 2 2 2 2 2

sin

cos

ln

1ln

1cos

1 sec

sen2 2

ln2 2

du u

aa uu

adu

u u au a

du u

au a u a a u

du a

a uu u au

a a

u a ua u du a u

a

u au a du u a u u a

= ∠−

= −∠

= + ±±

=± + ±

= ∠−

= ∠

− = − + ∠

± = ± ± + ±

∫

∫

∫

∫

∫

∫

MÁS INTEGRALES

( )

( )2 2

2 2

3

sin cossin

cos sincos

1 1sec sec tg ln sec tg

2 2

au

auau

e a bu b bue bu du

a b

e a bu b bue bu du

a b

u du u u u u

−=

++

=+

= + +

∫

∫

∫

au

ALGUNAS SERIES

( ) ( ) ( ) ( ) ( )( )

( ) ( )( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( )

2

0 00 0 0

0 0

2

2 3

3 5 7 2 11

2 4 6

'''

2!

: Taylor!

'' 00 ' 0

2!

0 : Maclaurin

!

12! 3! !

sin 13! 5! 7! 2 1 !

cos 12! 4!

nn

n n

nx

nn

f x x xf x f x f x x x

f x x x

n

f xf x f f x

f x

n

x x xe x

n

x x x xx x

n

x x xx

−−

−= + − +

−+ +

= + +

+ +

= + + + + + +

= − + − + + −−

= − + − ( ) ( )

( ) ( )

( )

2 21

2 3 41

3 5 7 2 11

16! 2 2 !

ln 1 12 3 4

tg 13 5 7 2 1

nn

nn

nn

x

n

x x x xx x

n

x x x xx x

n

−−

−

−−

+ + −−

+ = − + − + + −

∠ = − + − + + −−